1. State the problem: We want to find the value of $$X = \cos(57^\circ) \cos(27^\circ) + \sin(57^\circ) \sin(27^\circ)$$ given that \( \cos(57^\circ) = 0.5446 \), \( \cos(27^\circ) = 0.8910 \), and all angles are in degrees. 2. Recall the trigonometric identity: $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$ This matches the form of our expression exactly with \(A = 57^\circ\) and \(B = 27^\circ\). 3. Apply this identity to rewrite the expression: $$X = \cos(57^\circ - 27^\circ) = \cos(30^\circ)$$ 4. Evaluate \( \cos(30^\circ) \): We know \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660 \). 5. Conclusion: Therefore, the value of \( X \) is approximately: $$X \approx 0.8660$$ This agrees well with the given cosine values and uses the cosine difference identity to simplify the problem.